Lesson 26
1. Can Green’s third identity be directly used to find a solution to Laplace’s equation?
2. If is a solution to Laplace’s equation, why is n,n1 also a solution?
3. What is Green’s function? Write down the solution of Laplace’s equation in terms of Green’s function?
The function G in the above equation is a Green’s function.
equation.
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prevents This
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determine
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at the
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equation.
s Laplace'
of solutions
are ....
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then ˆ 0,
i.e.
solution,
a is
ˆ if constant,
are ts coefficien
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a is equation
s Laplace'
Since
2
2
n n
V y V
y V
V
y V
V
d G n dS
G G n
n dS G G n
d G
R
n dS G G n
R d G
V U
U
R R U
G ψ
y x
x y
y x y
y
y x
y
2 2
2 2
2 2
) (
) ( Therefore,
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4. How can the Green’s function be used to find solutions of Laplace’s equation?
5. What are spherical harmonics and why are they useful in finding solutions to Laplace’s equation?
In certain physical problems, the solution to Laplace’s equation is best expressed in terms of a series solution. For example, for the case of a spherical inclusion moving with a velocity u in an infinite fluid medium which is otherwise motionless, and the fluid is incompressible as well as irrotational, the resultant motion of the fluid can be found by solving Laplace’s equation.
6. What is Lengendre’s equation and when does it arise?
The solution to Laplace’s equation in a spherical domain can also be obtained by seeking a solution in the separated form:
This gives rise to Legendre’s equation and yields a series expansion in terms of spherical harmonics that involves Legendre’s polynomials.
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find can we , 0
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find can we If
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ndS G
n V G G
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y V
V
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x x
r r
r
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n
n 1
) (
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: solution series
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can then
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x x
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The n
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